Nuprl Lemma : det-fun+
∀[n:ℕ]. ∀[J:ℕn + 1]. ∀[r:Rng]. ∀[d:det-fun(r;n + 1)]. (λM.(d matrix+(r;J;M)) ∈ det-fun(r;n))
Proof
Definitions occuring in Statement :
matrix+: matrix+(r;j;M),
det-fun: det-fun(r;n),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
lambda: λx.A[x],
add: n + m,
natural_number: $n,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
det-fun: det-fun(r;n),
and: P ∧ Q,
nat: ℕ,
rng: Rng,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
false: False,
infix_ap: x f y,
so_lambda: λ2x y.t[x; y],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
so_apply: x[s1;s2],
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
subtract: n - m,
squash: ↓T,
true: True,
matrix+: matrix+(r;j;M),
matrix-mul-row: matrix-mul-row(r;k;i;M),
matrix-ap: M[i,j],
mx: matrix(M[x; y]),
less_than: a < b,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
matrix-swap-rows: matrix-swap-rows(M;i;j)
Lemmas referenced :
matrix+_wf,
matrix_wf,
rng_car_wf,
int_seg_wf,
istype-int,
set_subtype_base,
lelt_wf,
int_subtype_base,
istype-void,
matrix-swap-rows_wf,
matrix-mul-row_wf,
rng_times_wf,
mx_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
infix_ap_wf,
rng_plus_wf,
matrix-ap_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rng_minus_wf,
rng_zero_wf,
det-fun_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
rng_wf,
nat_wf,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-subtract-cancel,
decidable__lt,
less_than_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rng_sig_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rng_one_wf,
lt_int_wf,
assert_of_lt_int,
istype-top,
iff_weakening_uiff,
assert_wf,
rng_times_zero,
rng_plus_zero,
matrix_ap_mx_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
productElimination,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
universeIsType,
sqequalRule,
lambdaFormation_alt,
natural_numberEquality,
independent_pairFormation,
functionIsType,
inhabitedIsType,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality,
independent_isectElimination,
equalityIsType1,
productIsType,
unionElimination,
equalityElimination,
int_eqReduceTrueSq,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
equalityIsType2,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
int_eqReduceFalseSq,
addEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
hyp_replacement,
imageElimination,
universeEquality,
imageMemberEquality,
lessCases,
axiomSqEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[J:\mBbbN{}n + 1]. \mforall{}[r:Rng]. \mforall{}[d:det-fun(r;n + 1)]. (\mlambda{}M.(d matrix+(r;J;M)) \mmember{} det-fun(r;n))
Date html generated:
2019_10_16-AM-11_27_38
Last ObjectModification:
2018_10_10-PM-03_06_07
Theory : matrices
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